My attempt N.B To make this easier to read I'm skipping all the official statements of the induction process

Let
It is given that and positive so
Hence is less than

Assume true for

Test at

End of attempt

To finish this step confuses me. I implore the forum for assistance.

You're so close!

You have a nice way of writing 4-u_{k+1} in terms of u_k. So now you need to show that , which in turn shows that 4-u_{k+1}<4. This should be easy for you, since you have assumed that u_k < 4. Use that to show that the above fraction is < 4, thus 4-u_{k+1}<4, thus u_{k+1}<4. This shows that u_{k} < 4 implies u_{k+1}<4, which is the proof.